Small-eigenvalue asymptotics at the positive threshold

Let $a$ be a sufficiently regular kernel, and define

$$\kappa_0:=\lim_{\sigma\searrow\zeta}(\sigma-\zeta)^{-1/2}\widetilde{a}_\sigma(0),\qquad \kappa_2:=\lim_{\sigma\searrow\zeta}(\sigma-\zeta)^{-1}|\widetilde{a}_\sigma”(0)|.$$

Let $\xi_\sigma$ be the rescaled support parameter, $\overline{w}_\sigma$ the rescaled profile, $\overline{\chi}$ the indicator function of $\overline{I}$, and $u_\sigma$ the corresponding eigenfunction. Small-eigenvalue asymptotics at the positive threshold. One expects

$$\xi_\sigma\xrightarrow{\sigma\searrow0}\frac{3^{1/3}}{2^{1/3}\eta^{1/3}\kappa_2^{1/3}},\qquad (\sigma-\zeta)^{1/2}\overline{w}_\sigma(\overline{x})\xrightarrow{\sigma\searrow0}\frac{3^{2/3}\eta^{1/3}\theta\kappa_2^{1/3}}{2^{5/3}\kappa_0}\overline{\chi}(\overline{x})(1-\overline{x}^2).$$

Moreover,

$$\lim_{\sigma\to0}u_\sigma(x)=\theta,\qquad \lim_{\sigma\to0}\widetilde{f}(u_\sigma(x))=0$$

pointwise. This gives the limiting behavior in the regime where the kernel parameter approaches the positive threshold, and differs from the zero-threshold concentration regime.